A bound for the density of any Hausdorff space

Abstract

We show, in a certain specific sense, that both the density and the cardinality of a Hausdorff space are related to the "degree" to which the space is nonregular. It was shown by Sapirovskii that $d(X)\leq\pi\chi(X)^{c(X)}$ for a regular space $X$ and the author observed this holds if the space is only quasiregular. We generalize this result to the class of all Hausdorff spaces by introducing the nonquasiregularity degree $nq(X)$, which is countable when $X$ is quasiregular, and showing $d(X)\leq\pi\chi(X)^{c(X)nq(X)}$ for any Hausdorff space $X$. This demonstrates that the degree to which a space is nonquasiregular has a fundamental and direct connection to its density and, ultimately, its cardinality. Importantly, if $X$ is Hausdorff then $nq(X)$ is "small" in the sense that $nq(X)\leq\psi_c(X)$. This results in a unified proof of both Sapirovskii's density bound for regular spaces and Sun's bound $\pi\chi(X)^{c(X)\psi_c(X)}$ for the cardinality of a Hausdorff space $X$. A consequence is an improved bound for the cardinality of a Hausdorff space.